# Module #17. Work/Strain Hardening. READING LIST DIETER: Ch. 4, pp ; Ch. 6 in Dieter

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1 Module #17 Work/Strain Hardening READING LIST DIETER: Ch. 4, pp ; Ch. 6 in Dieter D. Kuhlmann-Wilsdorf, Trans. AIME, v. 224 (1962) pp

2 Work Hardening RECALL: During plastic deformation, dislocation density increases. We ve addressed this earlier. It is this increase in dislocation density that ultimately leads to work hardening. Dislocations interact with each other and can assume configurations that restrict the movement of other dislocations. The situation gets more severe as the dislocation density increases leading to an increase in the flow stress. [Ashby, Shercliff, & Cebon, p. 129] The dislocations can be either strong or weak obstacles depending upon the types of interactions that occurs between moving dislocations.

3 Work Hardening (2) To properly discuss work hardening, let s first consider the plastic deformation of a single crystal. F A o Plastic deformation is initiated at a critical stress, the critical resolved shear stress (CRSS). Recall from our derivation of the Taylor-Orowan equation that this is the same stress at which dislocations begin to move on a specific slip system. Slip plane normal A s F Slip direction coscos CRSS ys

4 Single crystal deformation (1) The characteristic vs. curve for a single crystal oriented for slip on one slip system (i.e., single slip) is shown to the right. CRSS Stage III Stage I Stage II It can be sub-divided into three stages based on work hardening behavior. Stage I: EASY GLIDE After yielding, shear stress for plastic deformation is almost constant. There is little or no work hardening. Typical of systems when single slip system is operative. Very few dislocation interactions: easy glide. Only interactions between the stress fields of dislocations to contend with. We mentioned this earlier. The active slip system has a maximum Schmid factor.

5 Single crystal deformation (2) Stage I: Conceptual description of easy glide Consider an array of dislocations moving on parallel slip planes. We ll use edge dislocations for schematic purposes. Edge dislocations have compressive stress fields above the slip plane and tensile stress fields below the slip plane. CRSS Stage I Stage III Stage II compression These stress fields interact during the early stages of deformation resulting in weak drag effects. Think of it as a friction force that inhibits glide. This frictional effect will increase as dislocation density increases. tension This is schematically illustrated to the right. You should consider how this relates to our prior discussions of dislocation configurations and the Peierls-Nabarro model for slip. Slip planes Attractive force In stage I dislocations are weak obstacles. Repulsive force

6 Single crystal deformation (3) Stage II: LINEAR HARDENING needed to continue plastic deformation begins to increase. This increase is approximately linear. Linear hardening. This stage begins when slip occurs on multiple slip systems. The work hardening rate increases due to interactions between dislocations moving on intersecting planes. This results in the production of jogs and other sessile dislocation configurations such as Lomer-Cottrell Locks, etc. CRSS Stage I Stage II More about Stage II on the next viewgraph Stage III Stage III: PARABOLIC HARDENING See a decreasing rate of work hardening due to an increase in the degree of cross slip. This is called parabolic hardening. Have a parabolic shape to the curve. NOTE: Polycrystals often go straight into stage III. Why?

7 Single crystal deformation (4) Stage II: Dislocation tangles ( forest dislocations ) form strong obstacles to dislocation motion. CRSS Stage I Stage II Stage III Tangles form when dislocation motion leads to the formation of immobile dislocation segments. Those segments will keep other dislocations from moving freely. Work hardening depends strongly on dislocation density: line length L unit volume L 1 L 3 2 The average separation distance between dislocations is: L constant Interactions can produce immobile dislocation configurations. Examples include Jogs and sessile dislocation locks (e.g., Lomer or Lomer-Cottrell locks).

8 Single crystal deformation (6) Recall the general hardening law : Stage III max Gb L CRSS Stage I Stage II If we substitute our expression for L into the equation above, we can express the shear flow stress as: o Gb where: o intrinsic flow strength for free material constant (0.2 for FCC, 0.4 for BCC) The strengthening increment derived from work hardening can now be approximated as: Gb or MGb k L constant G.I. Taylor, Proceedings of the Royal Society, A145, pp In this equation, M is the Taylor factor. G must be converted to E.

9 Single crystal deformation (7) Note for both materials (in this case Ti and Cu) that the stress ( and ) exhibits a ( ) ½ dependence Ti Cu FIGURE CRSS versus for Cu single crystals and polycrystals. - polycrystal; - single crystal, one slip system; - single crystal, two slip systems; - single crystal, six slip systems. [Image scanned from Courtney's text. Originally from H. Weidersich, J. Metals, 16 (1964) 425] FIGURE Flow stress versus for different grades of Ti deformed at room temperature. [R.L. Jones and H. Conrad, Trans. AIME, 245 (1969) ] Strength

10 Dislocation microstructure as function of p and Low strain: somewhat random. High strain: dense tangles and cellular arrangements. Form near the end of Stage II in single crystals. Cell boundaries have high. Cell interiors have low. Cells are called subgrains. Polycrystalline 304 Stainless steel Why do they form? Is strength still proportional to? Figure adapted from J.R. Foulds, A.M. Ermi, and J. Moteff, Materials Science and Engineering, 45 (1980) This journal paper provides a nice description of how dislocation microstructures evolve.

11 III II I CRSS Stage I Stage II Stage III I ~ G/10,000 This stage is often absent when multiple slip systems are operative II ~ G/300 Typical work hardening rates

12 Influence of crystal structure on single crystal stress-strain strain curves Shear Stress FCC (Cu) BCC (Nb) Stress-strain curves for three typical metallic crystal structures. Adapted from T. Suzuki, S. Takeuchi, and H. Yoshinaga, Dislocation Dynamics and Plasticity, Springer-Verlag, Berlin, 1991, p. 15. HCP (Mg) Shear Strain

13 In some crystal structures, dislocations dissociate to slip DOES THIS HAVE ANY IMPACT ON WORK HARDENING?

14 RECALL: Dissociation of unit dislocations in an FCC crystal In this example, the separation into partial dislocations is energetically favorable. There is a decrease in strain energy. A C B A ao 6 ao b ao 6 ao b ao 2 ao b C ao ao ao ao ao ao Separation produces a stacking fault between the partials. B

15 FCC Crystal STACKING FAULT Width = d SLIPPED A \$ a o b B C \$ \$ a o D b UNSLIPPED ao 2 ao b SFE d G Gb2b3 SFE 2 d partial dislocation separation shear modulus AB represents a regular (un-extended) dislocation. BC and BD represent partial dislocations. The region between BC and BC represents the stacking fault. In this region, the crystal has undergone intermediate slip. BC + stacking fault + BD represents an extended dislocation. Extended dislocations (in particular screw dislocations) define a specific slip plane. Thus, extended screw dislocations can only cross-slip when the partial dislocations recombine. See the illustration on the next page. This process requires some energy.

16 Extended Dislocation [Partial Dislocation + SF + Partial Dislocation] An extended screw dislocation must constrict before it can cross slip. Schematic illustration Cross-slip plane (11 1) plane (1) Extended (2) Formation of constricted segment (111) plane (4) (3) Cross-slip of constricted segment and separation into extended b (1) (2) Direction of motion (3) (4) Slip of extended on cross-slip plane

17 Influence of partial dislocations and SFE on deformation High SFE higher strain hardening rate Low SFE lower strain hardening rate Wavy slip lines Straight slip lines Dislocation cells and tangles Planar dislocation arrangement (a) Cu having a high SFE and therefore essentially un-dissociated dislocations. Cross-slip leads to the wavy slip bands and tangled dislocations (b) Cu-7.5 wt.% Al alloy, having a low SFE and dissociated dislocations. The absence of cross-slip leads to straight slip traces and uniform dislocation lines. Figure 9-11 Comparison of the deformation of two types of uniform polycrystalline materials in stage II deformation. From Guy, Introduction to Materials, McGraw-Hill, New York (1972) p. 419 [originally in Johnston and Feltner, Metallurgical Transactions, v. 1, n. 5 (1970) pp ].

18 Effect of γ SFE on Deformation Mechanism-1 (for FCC crystals) High γ SFE SFE SFE Gb b 2 d 2 3 Wavy slip steps observed Dislocation cells and tangles Pure Cu Few partial dislocations or small d. Thus, it is easier for cross-slip to occur. Pure Cu Random distribution of dislocations is observed because dislocations are not confined to specific slip planes. They can cross slip to overcome obstacles. Figure 9-11 Comparison of the deformation of two types of uniform polycrystalline materials in stage II deformation. From Guy, Introduction to Materials, McGraw-Hill, New York (1972) p. 419 [originally in Johnston and Feltner, Metallurgical Transactions, v. 1, n. 5 (1970) pp ].

19 Effect of γ SFE on Deformation Mechanism-2 (for FCC crystals) Low γ SFE SFE SFE Gb b 2 d 2 3 Straight slip steps observed Dislocations seem to be confined to specific bands/planes Cu 7.5 wt.% Al Slip is confined to specific planes. Leads to straight slip lines/steps on surface. Lots of partials, large d makes cross slip difficult. Cu 7.5 wt.% Al Dislocation movement is restricted due to because cross slip is inhibited. RECALL: partials must recombine to cross slip. [Images from Guy, Introduction to Materials, McGraw-Hill, New York (1972) p.419]

20 Partial to Partial Dislocation separation distance vs. γ SFE partial SF partial d Table: Stacking-fault free energies and separation between Shockley partial dislocations for metals (θ = 30 ). SFE 2 Gbp 2 2 cos2 1 8d 1 2 Metal γ (mj/m 3 ) a o (nm) b (nm) G (GPa) d (nm) Al Cu If θ = 0, ν = 1/3 Au Ni Gbp d or d 5 SFE 1 SFE Ag Notice the big change in d with γ SFE Cross-slip easier when d is small γ SFE is high

21 FCC Crystal SLIPPED A \$ It is more difficult to re-combine wide stacking faults (i.e., those with large d). Cross-slip is more difficult in materials with low SFE. Thus high SFE materials will work harden more rapidly. Material a o b B C ao 2 ao b SFE (mj/m 2 ) \$ \$ b Fault width Strain Hardening rate REASONS Stainless Steel <10 ~0.45 High Cross slip is more difficult Copper ~90 ~0.30 Med a o UNSLIPPED STACKING FAULT Width = d D Aluminum ~250 ~0.15 Low Cross slip is easier See pages in Hertzberg Gb2b3 SFE SFE 2 d d partial dislocation separation G shear modulus

22 Summary of hardening/strengthening mechanisms for crystalline solids Hardening Mechanism Nature of Obstacle Strong or Weak Hardening Law Work hardening Other dislocations Strong [1] Gb (see ) Grain size / Hall-Petch Grain boundaries Strong [2] ky d (see ) [3] 3/2 1/2 [4] Solid solution Solute atoms Weak (see ) Gs c / 700 (see ) [5] Deforming particles Small, coherent Weak (see ) 3/2 fr [6] particles CG (see ) b [7] Non-deforming particles Large, incoherent Strong (see ) Gb particles L 2r [1] equals about 0.2 for FCC metals and about 0.4 for BCC metals. [2] ky scales with inherent flow stress and/or shear modulus; therefore ky is generally greater for BCC metals than for FCC metals. [3] Exception to weak hardening occurs for interstitials in BCC metals; the shear distortion interacts with screw dislocations leading to strong hardening. [4] Equation apropos to substitutional atoms; parameter s is empirical, reflecting a combination of size and modulus hardening. [5] Coherent particles can be "strong" in optimally aged materials. [6] Constant C depends on specific mechanism of hardening; parameter relates to hardening mechanism(s). Equation shown applies to early stage precipitation. Late stage precipitation results in saturation hardening. [7] Highly overaged alloys can represent "weak" hardening. SYMBOLS : G shear modulus; b = Burgers vector; dislocation density ; d grain size; c solute atom concentration (at.%); f precipitate volume fraction; r precipitate radius; L spacing between precipitates on slip plane. Table adapted from Courtney, Mechanical Behavior of Materials, 2 nd edition, p. 232.

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